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The first and last terms of an A.P. are 18 and 48. If the sum of its terms is 396, then the number of terms will be?
10
15
9
12
12
Use the average value approach: Since the sum is the average of first and last terms multiplied by n, simply compute n=18+482├Ч396тАЛ=66792тАЛ=12.
First term (a) = 18, last term (l) = 48, and sum of terms (SnтАЛ) = 396.
SnтАЛ=2nтАЛ├Ч(a+l)
Use the average value approach: Since the sum is the average of first and last terms multiplied by n, simply compute n=18+482├Ч396тАЛ=66792тАЛ=12.
Students often confuse the sum formula SnтАЛ=2nтАЛ(a+l) with the nth term formula anтАЛ=a+(nтИТ1)d and waste time trying to find the common difference d which is not required.
Identify given variables
We are given the first term a=18, the last term l=48, and the sum SnтАЛ=396.
a=18,l=48,SnтАЛ=396
Select the appropriate summation formula
For an Arithmetic Progression (A.P.), when the first and last terms are known, the sum is given by SnтАЛ=2nтАЛ(a+l).
SnтАЛ=2nтАЛ(a+l)
Substitute and solve for n
Substitute the values: 396=2nтАЛ(18+48). Simplifying this, 396=2nтАЛ(66), which leads to 396=33n.
n=33396тАЛ=12
D is correct because solving the equation 396=2nтАЛ(18+48) yields n=12.
This concept of calculating the number of terms using the average of extremities is also frequently used in calculating the number of integers divisible by a certain value in a range.
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